nLab
double derivation

Given a commutativeringkk and an associative kk-algebra AA over kk, the tensor productA⊗kAA\otimes_k A is equipped with two bimodule structures, “outer” and “inner”. For the outer structure a⋅o(b⊗c)⋅od=ab⊗cda\cdot_o(b\otimes c)\cdot_o d = a b\otimes c d and for the inner a⋅i(b⊗c)⋅id=bd⊗aca\cdot_i(b\otimes c)\cdot_i d = b d\otimes a c. The two bimodule structures mutually commute. A kk-linear map α∈Homk(A,A⊗A)\alpha\in Hom_k(A,A\otimes A) is called a double derivation if it is also a map of AA-bimodules with respect to the outer bimodule structure (α∈AModA(AAA,AA⊗kAA)\alpha\in A Mod A({}_A A_A,{}_A A\otimes_k A_A)); thus the kk-module Der(A,A⊗A)Der(A,A\otimes A) of all double derivations becomes an AA-bimodule with respect to the innerAA-bimodule structure.

The tensor algebra TADer(A,A⊗A)T_A Der(A,A\otimes A) of the AA-bimodule Der(A,A⊗A)Der(A,A\otimes A) (which is the free monoid on Der(A,A⊗A)Der(A,A\otimes A) in the monoidal category of AA-bimodules) is a step in the definition of the deformed preprojective algebra?s of Bill Crawley-Boevey?. A theorem of van den Bergh says that for any associative AA the tensor algebra TADer(A,A⊗A)T_A Der(A,A\otimes A) has a canonical double Poisson bracket?.